Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Determination of the phase-difference distribution between two weak optical fields

Th. Richter

Published Oct 1, 1997• 3 citations
Publisher

Abstract

We reconsider the determination of the operational defined phase difference between two weak optical fields based on photon counting when the small number of possible experimental outcomes makes it harder to derive a continuous phase distribution. We show that the experimental procedure introduced to overcome this difficulty, namely, an external phase shifting of one input state, is not necessary. Indeed, all measuring data conditioned on the phase shift can be obtained from the data recorded without the external phase shift. Moreover, we investigate the effect of retaining ambiguous data or discarding even more data than only the ambiguous ones, respectively, and discuss the influence of a nonunit efficiency of the photodetectors.

DOI
10.1103/physreva.56.3134

Related papers

Sharpening the measured quantum phase distribution by discarding data

J. R. Torgerson, L. Mandeļ

In measurements of the phase difference between two weak coherent fields based on photon counting, it has been shown that discarding the outcomes with all-zero photon counts has the effect of sharpening the probability distribution of the phase difference. We show here that the distribution can be sharpened further by discarding even more data, such as those with fewer than two or three photon counts.

Optics Communications•Jan 1, 1997•7 citations

Measuring the Quantum Phase of the E.M. Field by Interference as ?0

J. R. Torgerson, L. Mandeļ

We show that the measured probability distribution of the phase difference between two electromagnetic fields can be sharpened progressively as data involving more and more detected photons are discarded, and that the phase difference can be well-defined even when the mean photon number n → 0. In our operational phase theory the variance of the directly measured phase difference has no lower bound other than zero, and this is consistent with the experimental results.

Physica Scripta•Jan 1, 1998•2 citations

Binomial states and the phase distribution measurement of weak optical fields

K. L. Pregnell, David Thomas Pegg

We show that the eight-port interferometer used by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 71, 2579 (1993)] to measure their operational phase distribution of light can also be used to measure the canonical phase distribution of weak optical fields, where canonical phase is defined as the complement of photon number. A binomial reference state is required for this purpose, which we show can be obtained to an excellent degree of approximation from a suitably squeezed state. The proposed method requires only photodetectors that can distinguish among zero photons, one photon, and more than one photon and is not particularly sensitive to photodetector imperfections.

arXiv:quant-ph/0508166•Physical Review A•Jun 24, 2003•18 citations

Operational approach to the phase of a quantum field

J. W. Noh, A. Fougères, L. Mandeļ

We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.

Physical Review A•Jan 1, 1992•237 citations

Accuracy of sampling quantum phase space in a photon counting experiment

Konrad Banaszek, Krzysztof Wódkiewicz

We study the accuracy of determining the phase space quasi-distribution of a single quantized light mode by a photon counting experiment. We derive an exact analytical formula for the error of the experimental outcome. This result provides an estimate of the experimental parameters, such as the number of events, required to determine the quasi-distribution with assumed precision. Our analysis also shows that it is, in general, not possible to compensate for the imperfection of the photodetector in the numerical processing of the experimental data. The discussion is illustrated with Monte Carlo simulations of the photon counting experiment for the coherent state, the one photon Fock state, and the Schrödinger cat state.

arXiv:physics/9610024•Journal of Modern Optics•Nov 1, 1997•12 citations

Direct and indirect strategies for phase measurement

Márcia T. Fontenelle, Samuel L. Braunstein, Wolfgang P. Schleich +1 more

Recently, Torgerson and Mandel [Phys. Rev. Lett. 76, 3939 (1996)] have reported a disagreement between two schemes for measuring the phase difference of a pair of optical fields. We analyze these schemes and derive their associated phase-difference probability distributions, including both their strong and weak field limits. Our calculation confirms the main point of Torgerson and Mandel of the non-uniqueness of an operational definition of the phase distribution. We further discuss the role of postselection of data and argue that it cannot meaningfully improve the sensitivity.

arXiv:quant-ph/9712032•Dec 16, 1997•1 citation
We reconsider the determination of the operational defined phase difference between two weak optical fields based on photon counting when the small number of possible experimental outcomes makes it harder to derive a continuous phase distribution. We show that the experimental procedure introduced to overcome this difficulty, namely, an external phase shifting of one input state, is not necessary. Indeed, all measuring data conditioned on the phase shift can be obtained from the data recorded without the external phase shift. Moreover, we investigate the effect of retaining ambiguous data or discarding even more data than only the ambiguous ones, respectively, and discuss the influence of a nonunit efficiency of the photodetectors.
We show that the eight-port interferometer used by Noh, Foug\`eres, and Mandel [Phys. Rev. Lett. 71, 2579 (1993)] to measure their operational phase distribution of light can also be used to measure the canonical phase distribution of weak optical fields, where canonical phase is defined as the complement of photon number. A binomial reference state is required for this purpose, which we show can be obtained to an excellent degree of approximation from a suitably squeezed state. The proposed method requires only photodetectors that can distinguish among zero photons, one photon, and more than one photon and is not particularly sensitive to photodetector imperfections.
We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.
We study the accuracy of determining the phase space quasi-distribution of a single quantized light mode by a photon counting experiment. We derive an exact analytical formula for the error of the experimental outcome. This result provides an estimate of the experimental parameters, such as the number of events, required to determine the quasi-distribution with assumed precision. Our analysis also shows that it is, in general, not possible to compensate for the imperfection of the photodetector in the numerical processing of the experimental data. The discussion is illustrated with Monte Carlo simulations of the photon counting experiment for the coherent state, the one photon Fock state, and the Schrödinger cat state.
Recently, Torgerson and Mandel [Phys. Rev. Lett. 76, 3939 (1996)] have reported a disagreement between two schemes for measuring the phase difference of a pair of optical fields. We analyze these schemes and derive their associated phase-difference probability distributions, including both their strong and weak field limits. Our calculation confirms the main point of Torgerson and Mandel of the non-uniqueness of an operational definition of the phase distribution. We further discuss the role of postselection of data and argue that it cannot meaningfully improve the sensitivity.